On the /spl Nscr//spl Pscr/-hardness of the purely complex μ computation, analysis/synthesis, and some related problems in multidimensional systems

Abstract

It is shown that for a given complex matrix M, and a purely complex uncertainty structure /spl Delta/, the problem of checking whether the inequality /spl mu//sub /spl Delta//(M)<1 holds, is /spl Nscr//spl Pscr/-hard. It is also shown that, the problem of checking whether the frequency domain /spl mu/-norm, /spl par/M(s)/spl par//sub /spl mu//, of an LTI system, M(s), is less than 1, and the problem of checking whether the best achievable /spl mu/-norm, inf/sub Q/spl epsiv//spl Hscr//spl infin///spl par//spl Fscr/(T,Q)/spl par//sub /spl mu//, of an LFT, /spl Fscr/(T,Q), is less than one, are both /spl Nscr//spl Pscr/-hard problems, namely purely complex /spl mu/ computation, analysis/synthesis are all /spl Nscr//spl Pscr/-hard. Although general /spl Hscr//sup /spl infin// norm computation, analysis/synthesis have a well established theory for LTI systems, there is no known nonconservative polynomial time procedure for purely complex /spl mu/ computation, analysis/synthesis problems. The results obtained imply that it is rather unlikely to find nonconservative polynomial time procedures for the purely complex /spl mu/ computation, analysis/synthesis problem, contrary to the standard /spl Hscr//sup /spl infin// problems. As independent results, it is also shown that the problem of checking the stability and the problem of computing the /spl Hscr//sup /spl infin// norm, are both /spl Nscr//spl Pscr/-hard problems for multidimensional systems. These results imply that it is rather unlikely to find a simple analogue of the Schur-Cohn test for checking the stability and an efficient generalization of bisection method for computing the /spl Hscr//sup /spl infin// norm, in the context of multidimensional systems.